Multi-critical unitary random matrix ensembles and the general Painlevé II equation

We study unitary random matrix ensembles of the form Z-'N\detM\2«e-NTrVWdM, where a > -1/2 and V is such that the limiting mean eigenvalue density for n, N - > oc and n/N - ► 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight \x\2ae~NV^x\ Here the main focus is on the construction of a local parametrix near the origin with ^-functions associated with a special solution qa of the Painleve II equation q" = sq + 2q3 - a. We show that qa has no real poles for a > -1/2, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit.

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